Science:Math Exam Resources/Courses/MATH105/April 2015/Question 05 (a)
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Question 05 (a) 

Suppose that and Find the Maclaurin series for 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

Expand as a Maclaurin series by using the geometric series 
Hint 2 

Once a series for is obtained, integrate it termwise and use the condition to obtain a series for . 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Since , a Maclaurin series for can be computed from the geometric series By substituting , we obtain: To obtain a series for , we integrate the series for term by term and apply the initial condition . Integrating the series for implies that for some constant C determined by the initial condition. Since evaluates to 0 at , we must have for . Therefore, a Maclaurin series for is given by 